Show that the point of intersection Q of the axis of the parabola y=x^ 2
and the hypotenuse of right triangle RST (inscribed in the parabola so
that R coincides with the vertex of the parabola) is independent of the
choice of right triangle.

From a point P on the circumcircle of the triangle ABC perpendiculars
are dropped to the sides AB, BC, CA. Prove that the line joining the
feet of the perpendiculars bisects the line joining the orthocentre of
triangle ABC and point P.

Is it possible to have a triangle with two 90 degree angles, where the
other two legs from the connected 90 degree angles meet to finish the
triangle? Where would you find such a triangle? I thought it might
work if the triangle is on a sphere, but then the lines aren't straight.